// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_SVDBASE_H
#define EIGEN_SVDBASE_H

namespace Eigen {

namespace internal {
template<typename Derived>
struct traits<SVDBase<Derived>> : traits<Derived>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};
}

/** \ingroup SVD_Module
 *
 *
 * \class SVDBase
 *
 * \brief Base class of SVD algorithms
 *
 * \tparam Derived the type of the actual SVD decomposition
 *
 * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
 *   \f[ A = U S V^* \f]
 * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero
 * outside of its main diagonal; the diagonal entries of S are known as the \em singular \em values of \a A and the
 * columns of \a U and \a V are known as the left and right \em singular \em vectors of \a A respectively.
 *
 * Singular values are always sorted in decreasing order.
 *
 *
 * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p
 * matrix, letting \a m be the smaller value among \a n and \a p, there are only \a m singular vectors; the remaining
 * columns of \a U and \a V do not correspond to actual singular vectors. Asking for \em thin \a U or \a V means asking
 * for only their \a m first columns to be formed. So \a U is then a n-by-m matrix, and \a V is then a p-by-m matrix.
 * Notice that thin \a U and \a V are all you need for (least squares) solving.
 *
 * The status of the computation can be retrived using the \a info() method. Unless \a info() returns \a Success, the
 * results should be not considered well defined.
 *
 * If the input matrix has inf or nan coefficients, the result of the computation is undefined, and \a info() will
 * return \a InvalidInput, but the computation is guaranteed to terminate in finite (and reasonable) time. \sa class
 * BDCSVD, class JacobiSVD
 */
template<typename Derived>
class SVDBase : public SolverBase<SVDBase<Derived>>
{
  public:
	template<typename Derived_>
	friend struct internal::solve_assertion;

	typedef typename internal::traits<Derived>::MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	typedef typename Eigen::internal::traits<SVDBase>::StorageIndex StorageIndex;
	typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
	enum
	{
		RowsAtCompileTime = MatrixType::RowsAtCompileTime,
		ColsAtCompileTime = MatrixType::ColsAtCompileTime,
		DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
		MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
		MatrixOptions = MatrixType::Options
	};

	typedef Matrix<Scalar,
				   RowsAtCompileTime,
				   RowsAtCompileTime,
				   MatrixOptions,
				   MaxRowsAtCompileTime,
				   MaxRowsAtCompileTime>
		MatrixUType;
	typedef Matrix<Scalar,
				   ColsAtCompileTime,
				   ColsAtCompileTime,
				   MatrixOptions,
				   MaxColsAtCompileTime,
				   MaxColsAtCompileTime>
		MatrixVType;
	typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;

	Derived& derived() { return *static_cast<Derived*>(this); }
	const Derived& derived() const { return *static_cast<const Derived*>(this); }

	/** \returns the \a U matrix.
	 *
	 * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
	 * the U matrix is n-by-n if you asked for \link Eigen::ComputeFullU ComputeFullU \endlink, and is n-by-m if you
	 * asked for \link Eigen::ComputeThinU ComputeThinU \endlink.
	 *
	 * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
	 *
	 * This method asserts that you asked for \a U to be computed.
	 */
	const MatrixUType& matrixU() const
	{
		_check_compute_assertions();
		eigen_assert(computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
		return m_matrixU;
	}

	/** \returns the \a V matrix.
	 *
	 * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
	 * the V matrix is p-by-p if you asked for \link Eigen::ComputeFullV ComputeFullV \endlink, and is p-by-m if you
	 * asked for \link Eigen::ComputeThinV ComputeThinV \endlink.
	 *
	 * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
	 *
	 * This method asserts that you asked for \a V to be computed.
	 */
	const MatrixVType& matrixV() const
	{
		_check_compute_assertions();
		eigen_assert(computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
		return m_matrixV;
	}

	/** \returns the vector of singular values.
	 *
	 * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
	 * returned vector has size \a m.  Singular values are always sorted in decreasing order.
	 */
	const SingularValuesType& singularValues() const
	{
		_check_compute_assertions();
		return m_singularValues;
	}

	/** \returns the number of singular values that are not exactly 0 */
	Index nonzeroSingularValues() const
	{
		_check_compute_assertions();
		return m_nonzeroSingularValues;
	}

	/** \returns the rank of the matrix of which \c *this is the SVD.
	 *
	 * \note This method has to determine which singular values should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index rank() const
	{
		using std::abs;
		_check_compute_assertions();
		if (m_singularValues.size() == 0)
			return 0;
		RealScalar premultiplied_threshold =
			numext::maxi<RealScalar>(m_singularValues.coeff(0) * threshold(), (std::numeric_limits<RealScalar>::min)());
		Index i = m_nonzeroSingularValues - 1;
		while (i >= 0 && m_singularValues.coeff(i) < premultiplied_threshold)
			--i;
		return i + 1;
	}

	/** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(),
	 * which need to determine when singular values are to be considered nonzero.
	 * This is not used for the SVD decomposition itself.
	 *
	 * When it needs to get the threshold value, Eigen calls threshold().
	 * The default is \c NumTraits<Scalar>::epsilon()
	 *
	 * \param threshold The new value to use as the threshold.
	 *
	 * A singular value will be considered nonzero if its value is strictly greater than
	 *  \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$.
	 *
	 * If you want to come back to the default behavior, call setThreshold(Default_t)
	 */
	Derived& setThreshold(const RealScalar& threshold)
	{
		m_usePrescribedThreshold = true;
		m_prescribedThreshold = threshold;
		return derived();
	}

	/** Allows to come back to the default behavior, letting Eigen use its default formula for
	 * determining the threshold.
	 *
	 * You should pass the special object Eigen::Default as parameter here.
	 * \code svd.setThreshold(Eigen::Default); \endcode
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	Derived& setThreshold(Default_t)
	{
		m_usePrescribedThreshold = false;
		return derived();
	}

	/** Returns the threshold that will be used by certain methods such as rank().
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	RealScalar threshold() const
	{
		eigen_assert(m_isInitialized || m_usePrescribedThreshold);
		// this temporary is needed to workaround a MSVC issue
		Index diagSize = (std::max<Index>)(1, m_diagSize);
		return m_usePrescribedThreshold ? m_prescribedThreshold : RealScalar(diagSize) * NumTraits<Scalar>::epsilon();
	}

	/** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
	inline bool computeU() const { return m_computeFullU || m_computeThinU; }
	/** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
	inline bool computeV() const { return m_computeFullV || m_computeThinV; }

	inline Index rows() const { return m_rows; }
	inline Index cols() const { return m_cols; }

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
	 *
	 * \param b the right-hand-side of the equation to solve.
	 *
	 * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
	 *
	 * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and
	 * least-squares solving. In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$
	 * \Vert A x - b \Vert \f$.
	 */
	template<typename Rhs>
	inline const Solve<Derived, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	/** \brief Reports whether previous computation was successful.
	 *
	 * \returns \c Success if computation was successful.
	 */
	EIGEN_DEVICE_FUNC
	ComputationInfo info() const
	{
		eigen_assert(m_isInitialized && "SVD is not initialized.");
		return m_info;
	}

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	void _check_compute_assertions() const { eigen_assert(m_isInitialized && "SVD is not initialized."); }

	template<bool Transpose_, typename Rhs>
	void _check_solve_assertion(const Rhs& b) const
	{
		EIGEN_ONLY_USED_FOR_DEBUG(b);
		_check_compute_assertions();
		eigen_assert(computeU() && computeV() &&
					 "SVDBase::solve(): Both unitaries U and V are required to be computed (thin unitaries suffice).");
		eigen_assert((Transpose_ ? cols() : rows()) == b.rows() &&
					 "SVDBase::solve(): invalid number of rows of the right hand side matrix b");
	}

	// return true if already allocated
	bool allocate(Index rows, Index cols, unsigned int computationOptions);

	MatrixUType m_matrixU;
	MatrixVType m_matrixV;
	SingularValuesType m_singularValues;
	ComputationInfo m_info;
	bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
	bool m_computeFullU, m_computeThinU;
	bool m_computeFullV, m_computeThinV;
	unsigned int m_computationOptions;
	Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
	RealScalar m_prescribedThreshold;

	/** \brief Default Constructor.
	 *
	 * Default constructor of SVDBase
	 */
	SVDBase()
		: m_info(Success)
		, m_isInitialized(false)
		, m_isAllocated(false)
		, m_usePrescribedThreshold(false)
		, m_computeFullU(false)
		, m_computeThinU(false)
		, m_computeFullV(false)
		, m_computeThinV(false)
		, m_computationOptions(0)
		, m_rows(-1)
		, m_cols(-1)
		, m_diagSize(0)
	{
		check_template_parameters();
	}
};

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename Derived>
template<typename RhsType, typename DstType>
void
SVDBase<Derived>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	// A = U S V^*
	// So A^{-1} = V S^{-1} U^*

	Matrix<typename RhsType::Scalar,
		   Dynamic,
		   RhsType::ColsAtCompileTime,
		   0,
		   MatrixType::MaxRowsAtCompileTime,
		   RhsType::MaxColsAtCompileTime>
		tmp;
	Index l_rank = rank();
	tmp.noalias() = m_matrixU.leftCols(l_rank).adjoint() * rhs;
	tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
	dst = m_matrixV.leftCols(l_rank) * tmp;
}

template<typename Derived>
template<bool Conjugate, typename RhsType, typename DstType>
void
SVDBase<Derived>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	// A = U S V^*
	// So  A^{-*} = U S^{-1} V^*
	// And A^{-T} = U_conj S^{-1} V^T
	Matrix<typename RhsType::Scalar,
		   Dynamic,
		   RhsType::ColsAtCompileTime,
		   0,
		   MatrixType::MaxRowsAtCompileTime,
		   RhsType::MaxColsAtCompileTime>
		tmp;
	Index l_rank = rank();

	tmp.noalias() = m_matrixV.leftCols(l_rank).transpose().template conjugateIf<Conjugate>() * rhs;
	tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
	dst = m_matrixU.template conjugateIf<!Conjugate>().leftCols(l_rank) * tmp;
}
#endif

template<typename MatrixType>
bool
SVDBase<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
{
	eigen_assert(rows >= 0 && cols >= 0);

	if (m_isAllocated && rows == m_rows && cols == m_cols && computationOptions == m_computationOptions) {
		return true;
	}

	m_rows = rows;
	m_cols = cols;
	m_info = Success;
	m_isInitialized = false;
	m_isAllocated = true;
	m_computationOptions = computationOptions;
	m_computeFullU = (computationOptions & ComputeFullU) != 0;
	m_computeThinU = (computationOptions & ComputeThinU) != 0;
	m_computeFullV = (computationOptions & ComputeFullV) != 0;
	m_computeThinV = (computationOptions & ComputeThinV) != 0;
	eigen_assert(!(m_computeFullU && m_computeThinU) && "SVDBase: you can't ask for both full and thin U");
	eigen_assert(!(m_computeFullV && m_computeThinV) && "SVDBase: you can't ask for both full and thin V");
	eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime == Dynamic) &&
				 "SVDBase: thin U and V are only available when your matrix has a dynamic number of columns.");

	m_diagSize = (std::min)(m_rows, m_cols);
	m_singularValues.resize(m_diagSize);
	if (RowsAtCompileTime == Dynamic)
		m_matrixU.resize(m_rows, m_computeFullU ? m_rows : m_computeThinU ? m_diagSize : 0);
	if (ColsAtCompileTime == Dynamic)
		m_matrixV.resize(m_cols, m_computeFullV ? m_cols : m_computeThinV ? m_diagSize : 0);

	return false;
}

} // end namespace

#endif // EIGEN_SVDBASE_H
